An algebra problem by Phudish Prateepamornkul

Algebra Level 4

\[\begin{cases} a^{3} b^{2}+b^{3} c^{2}+c^{3}a^{2} = \dfrac{103}{1024} \\ a^{2} b^{3}+b^{2}c^{3}+c^{2}a^{3} = \dfrac{125}{1024} \\ a^{2} b^{2} c+a^{2} b c^{2}+a b^{2}c^{2} = \dfrac{33}{512}\end{cases} \]

Let reals \(a\),\(b\) and \(c\) satisfy the system of equations above. If \(\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b} = - \dfrac{x}{y}\), where \(x\) and \(y\) are coprime positive integers, find \(x+y\).

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